The invention relates to a method for learning systems, and in particular, to learning systems that generate rules by derivation from a logical theory.
Several methods for rule induction are known in the art. Common for these methods is that a set of logical rules are generated from a set of examples, where each example has been given a label, which can either be a categorical value or a numeric value. Each logical rule consists of a condition part that in turn consists of a set of logical tests as well as a conclusion part, which assigns a value for the label. For example, the condition part may be that the number of atoms must exceed the numerical value five and the molecule weight must not be less than 100 to generate a positive class. The examples may include attributes, such as molecule weight, that correspond to the condition part of the logical rule. One of the most common techniques for rule induction is known as decision tree induction, which generates a set of hierarchically organized rules, where none of the rules overlap (i.e., the conditions of two different rules are mutually exclusive). Examples of such techniques are ID3 and CART. Other techniques, such as covering or separate-and-conquer, may generate overlapping rules. Examples of such techniques are CN2 and RIPPER.
Most techniques for rule induction allow examples to be represented as fixed-length attribute-value vectors, and the conditions to consist of simple tests that, for example, checks whether a particular attribute has a particular value, or whether the value is below or above a particular threshold. Some techniques also allow examples to be represented by arbitrary logical terms, including lists and trees, and conditions to consist of arbitrary logical literals such as tests that involve an arbitrary number of variables using arbitrarily defined predicates. Such techniques are studied primarily in a research field known as inductive logic programming and examples of such techniques are FOIL and PROGOL.
One method for rule induction is to use a logical theory from which rules are derived by using an inference procedure known as resolution. During the generation of rules to be included in the final hypothesis, a large number of candidate rules are evaluated, which involves checking for a set of training examples, which of these fulfill the conditions of the candidate rules. After the final hypothesis has been generated, it is usually applied to examples not included in the set of training examples, which again involves checking whether or not the conditions of the rules are fulfilled for each example. This is a very cumbersome process, in which complex proof trees may have to be generated repeatedly for each example. In both cases, minimizing the amount of time required to perform these tests can be of high importance. There is a need for a more effective process that does not require the repeated generation of proof trees for the examples.
The present invention provides a solution to this problem and is a method and an apparatus for efficiently checking whether or not the conditions of a rule derived by resolution from a logical theory are fulfilled by an example. The apparatus consists of the following three modules:
i) A module for generating a database from proof trees that have been constructed from the examples using the logical theory;
ii) A module for generating database queries from rules that have been derived from the logical theory; and
iii) A module for querying the database with the queries obtained from the rules.
More particularly, the method is used in a computer and includes the steps of providing a logical theory that has clauses. A rule is generated that is a resolvent of clauses in the logical theory. An example is retrieved. A proof tree is generated from the example using the logical theory. The proof tree is transformed into a database of a coverage check apparatus. The rule is converted into a partial proof tree that has nodes. The partial proof tree is transformed into a database query of the coverage check apparatus. The query is executed to identify tuples in the database that correspond to the nodes of the partial proof tree. In this way, the database of pre-existing examples may be investigated to determine if a rule covers a pre-existing example, that are associated with the same logical theory, so there is no need to recreate complicated proof trees for the examples.